


Nonlinear Schrodinger Equations
Org: James Colliander and Robert Jerrard (Toronto) [PDF]
 IOAN BEJENARU, UCLA
Global solutions for Schroedinger Maps
[PDF] 
We discuss the global theory for Schrödinger Maps in higher
dimensions. For n ³ 3 we establish the global wellposedness for
small initial data in the critical Besov space, while for n ³ 4
we establish the global wellposedness for small initial data in the
critical Sobolev space.
 PIETER BLUE, University of Toronto, Toronto, ON, M5S 2E4
Decay of waves on a warp product manifold with an unstable,
closed geodesic surface
[PDF] 
In this talk we compare the decay of solutions to the wave equation in
R^{3} to those of the wave equation on a threedimensional warp
product space, (x,w) Î R ×S^{2} with metric dx^{2} +r(x)^{2} dw^{2}. If the radii of spheres, r(x), has a unique
minimum, then the sphere of minimal radius is a closed geodesic
surface. Heuristically, this should impede the decay of waves, since
waves follow the paths of geodesics. Indeed, Ralston has shown that
initial data can be chosen for which an arbitrarily large percentage
of the energy (H^{1} density) remains within a neighbourhood of the
geodesic for arbitrary long periods of time. Thus, if decay estimates
hold, there must be some loss of regularity, with higher derivatives
used to control the decay of localised energy. The conformal charge
is a weighted H^{1} norm defined by analogy to R^{n}, and its growth
is controlled by the time integral of the energy near the geodesic
surface. Using refinements of previous local decay arguments, the
conformal charge can be shown to be bounded with an epsilon loss of
regularity if the geodesic surface is unstable. This gives the same
rate of decay for certain L^{p} norms as in R^{n}. This argument uses
vector field techniques and can be extended to small data, nonlinear
problems under additional assumptions on the growth of r(x). Since
vector field techniques are analogous to commutator methods, we expect
that similar methods will apply to the NLS on manifolds.
 DANIELA DE SILVA, Johns Hopkins University
Low regularity solutions for a 2D quadratic nonlinear
Schrodinger equation
[PDF] 
In this talk, we will discuss the initial value problem for the
quadratic nonlinear Schrödinger equation
where u : R^{2} ×R ® C. In a
recent work in collaboration with I. Bejenaru, we proved that this
problem is locally wellposed in H^{s} (R^{2}) when s > 1.
The critical exponent for this problem is s_{c} = 1, and previous
work of J. Colliander, J. Delort, C. Kenig and G. Staffilani,
established local wellposedness for s > 3/4.
 ROY GOODMAN, New Jersey Institute of Technology, Newark, NJ, USA
Fractal Structure in Solitary Wave Interactions
[PDF] 
The following scenario has been seen in many nonintegrable,
dispersive, nonlinear PDE over the last 25 years: two solitary waves
are propagated on a collision course. Above some critical velocity
v_{c}, they simply bounce off each other. Below
v_{c} they may be captured and merge into a single
localized mass, or they may interact a finite number of times before
escaping each other's embrace. Whether they are captured, and how
many times the solitary waves interact before escape, depends on the
initial velocity in a complicated manner, often remarked, though never
shown, to be a fractal (a chaotic scattering process). This has been
observed in coupled NLS, sineGordon, phi^{4}, and others.
These PDE systems are commonly studied by (nonrigorously) deriving a
reduced set of ODE that numerically reproduce this behavior. Using
matched asymptotics and Melnikov integrals, we give asymptotic
formulas for v_{c} and for certain salient features of the
fractal structure. We derive a discretetime iterated map through
which the entire structure can be unravelled.
Joint with Richard Haberman, Southern Methodist University.
 MANOUSSOS GRILLAKIS, Maryland

 NATASA PAVLOVIC, Princeton University, Department of Mathematics, Princeton,
NJ 085441000, USA
Global wellposedness for the L^{2}critical NLS in higher
dimensions
[PDF] 
In this talk we will present a joint work with Daniela De Silva,
Gigliola Staffilani and Nikolaos Tzirakis on global wellposedness for
the L^{2} critical NLS in R^{n} with n ³ 3. Inspired
by a recent paper of Fang and Grillakis, we combine the method of
almost conservation laws with a local in time Morawetz estimate to
improve global wellposedness results in higher dimensions.
 ZOI RAPTI, University of Illinois at UrbanaChampaign
Modulational Instability for NLS equations with a periodic
potential
[PDF] 
This talk is about the stability properties of solutions to the NLS
equation with a periodic potential which bifurcate from the
FloquetBloch eigenstates of the linear problem in the small
amplitude limit. We exploit the symmetries of the problem, in
particular the fact that the monodromy matrix is a symplectic matrix.
We find that the solutions corresponding to the band edges alternate
stability, with the first band edge being modulationally unstable in
the focusing case, the second band edge being modulationally unstable
in the defocusing case, and so on.
 LENYA RYZHIK, University of Chicago
Diffusion and flow mixing
[PDF] 
I will describe some recent results on the interplay between diffusion
and strong advection by an incompressible flow.
 NIKOS TZIRAKIS, University of Toronto
Morawetz type inequalities and improved global
wellposedness for the quintic NLS in 1d
[PDF] 
Morawetz type estimates are monotonicity formulae that take advantage
of the momentum conservation law of the nonlinear Schrödinger
equation (NLS), and have been used extensively in obtaining global
wellposedness and scattering results. By using an interaction
Morawetz inequality for an "almost solution" of the NLS we prove a
localintime L_{t}^{6} L_{x}^{6} bound. We use this bound along with the
"Imethod" to prove a new global wellposedness result for the
quintic NLS in 1d.
This is joint work with D. De Silva, N. Pavlovic and G. Staffilani.
 MONICA VISAN, Institute for Advanced Study, One Einstein Drive, Princeton,
NJ 08540
The masscritical NLS
[PDF] 
We discuss recent progress on global wellposedness and scattering for
the masscritical NLS with initial data in L^{2}.
The results presented are joint work with Terence Tao and Xiaoyi
Zhang.
 JIE XIAO, Memorial University, St. John's, NL, A1C 5S7
Old and New Morrey Spaces with Heat Kernel Bounds
[PDF] 
In this talk I will address my work joint with D. Xuong and L. Yan.
More precisely, given p Î [1,¥) and l Î (0,n), we
discuss Morrey space L^{p,l} (R^{n}) of all locally
integrable complexvalued functions f on R^{n} such that
for every open Euclidean ball B Ì R^{n} with radius
r_{B} there are numbers C=C(f) (depending on f) and c=c(f,B)
(relying upon f and B) satisfying
r_{B}^{l} 
ó õ

B

f(x)c^{p} dx £ C 

and derive old and new, two essentially different cases arising from
either choosing c = f_{B} = B^{1} ò_{B} f(y) dy or replacing c
by P_{tB}(x) = ò_{tB} p_{tB} (x,y) f(y) dywhere t_{B} is
scaled to r_{B} and p_{t} (·,·) is the kernel of the
infinitesimal generator L (taking the Schroedinger operator as a
special one) of an analytic semigroup {e^{tL}}_{t ³ 0} on
L^{2} (R^{n}). Consequently, we are led to simultaneously
characterize the old and new Morrey spaces, but also to show that for
a suitable operator L, the new Morrey space is equivalent to the old
one.
 VADIM ZHARNITSKY, University of Illinois
Dispersion managed NLS and Strichartz inequality
[PDF] 
Dispersion managed NLS is a model for optical pulse propagation in a
fiber with piecewise constant dispersion. In a certain parameter
regime, this equation possesses approximately periodic solitary waves
called dispersion managed solitons. They turn out to be minimizers of
an averaged variational principle. This variational principle is
closely related to the Strichartz inequality. We will describe some
interplay between these objects and in particular, we will present an
easy proof of the classical version of the Strichartz inequality.
This is joint work with Dirk Hundertmark.
 GANG ZHOU, University of Toronto
Perturbation Expansion and Nth Order Fermi Golden Rule of
the Nonlinear Schrödinger Equations
[PDF] 
In this presentation I will show the asymptotic stability of trapped
solitions of generalized nonlinear Schrödinger equations with
external potentials. We use Fermi Golden rule (FGR) to show the
dynamics of the soliton which is close to Newton's equation plus a
radiation term. We compute the expressions for the fourth and the
sixth order Fermi Golden Rules (FGR) by perturbation expansion.

